A STUDY OF J-INTEGRAL OF THE ORTHOTROPIC COMPOSITE MATERIAL

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Use of photoelasticity to determine orthotropicK I stress-intensity factor

A new experimental method of obtaining orthotropic stress-intensity factor,K _I , is presented. The orthotropic photoelasticity and orthotropic linear-elastic fracture-mechanics laws are combined. The combined set of equations is used along with half-fringe photoelasticity to determineK _I in a compact-tension specimen made of a transparent unidirectional fiberglass-epoxy material. The results are compared with finite-element-method solutions.     

Embedded fine-grid method applied to three-dimensional stress-intensity-factor analysis

The purpose of this paper is to extend the embedded fine-grid method to three-dimensional stress-intensity-factor analysis. The embedded crossed fine grids give two components of displacement and are capable of calculatingK _I without the assumption of the plane-strain condition along a crack or a notch front. It is not valid to assume the plane-strain condition to calculateK _I in the vicinity of a free surface, whereK _I is influenced by a free surface (plane-stress condition). In this paper, a dyeing and bleaching process is considered to reproduce the crossed fine grids on an epoxy plate. By using these grids as embedded grids, the distributions ofK _I along notch fronts in SEN specimens with various different thicknesses and side grooves are studied. The influences of a free surface and the side-groove effects on the distribution ofK _I are discussed. Paper was presented at 1982 SESA/JSME Spring Meeting held in Oahu and Maui, HI on May 24–29.     

Associated path independent J-integrals for separating mixed modes

The displacement field in two dimensional elasticity is separated into symmetrical and antisymmetrical parts u_i = u^I_i + u^II_i. These components are new elastostatic fields and lead to two path independent integrals of the J-integral type. The integrals J_I J_II are respectively the symmetrical and antisymmetrical components of the J-integral of Rice: J = J_I + J_II. Use of the associated path independent integrals allows K_I, and K_II to be calculated separately.     

An experimental analysis of displacement field of mixed mode crack

In this paper, the characteristic properties ofv (y-direction displacement) field surrounding the tip of a mixed mode crack are studied. These properties can be used to evaluate the rigid body rotation of the crack tip, theK _I SIF and the ratio ofK _II SIF toK _I.The authors employ a film to record the displacement information based on the technique of moire interferometry with sticking films. By using the data taken from the moire pattern and treating them with the damping least square method, all of the parameters of the crack can be obtained accurately.     

OnJ-integrals in the plane fracture of composite materials

By using a complex function method in this paper, the complex form of J-integral of mixed mode crack tip for unidirectional plate of linear-elastic orthotropic composites is obtained first by substituting crack tip stresses and displacements into general formula of J-integral. And then, the path-independence of this J-integral is proved. Finally, the computing formula of this J-integral is derived. As special examples, the complex forms, path-independence and computing formulae of J-integrals of mode Ⅰ and mode Ⅱ crack tips for unidirectional plate of linear-elastic orthotropic composites are given.     

A method for measuring mode I crack tip constraint under static and dynamic loading conditions

A novel experimental technique for measuring crack tipT-stress, and hence in-plane crack tip constraint, in elastic materials has been developed. The method exploits optimal positioning of stacked strain gage rosette near a mode I crack tip such that the influence of dominant singular strains is negated in order to determineT-stress accurately. The method is demonstrated for quasi-static and low-velocity impact loading conditions and two values of crack length to plate width ratios (a/W). By coupling this new method with the Dally-Sanford single strain gage method for measuring the mode I stress intensity factorK _I , the crack tip biaxiality parameter is also measured experimentally. Complementary small strain, static and dynamic finite element simulations are carried out under plane stress conditions. Time histories ofK _I andT-stress are computed by regression analysis of the displacement and stress fields, respectively. The experimental results are in good agreement with those obtained from numerical simulations. Preliminary data for critical values ofK _I and β for dynamic experiments involving epoxy specimens are reported. Dynamic crack initiation toughness shows an increasing trend as β becomes more negative at higher impact velocities.     

Numerical simulation of stress intensity factors via the thermoelastic technique

The purpose of this study is to investigate the accuracy of the least squares method for finding the in-plane stress intensity factorsK _I andK _II using thermoelastic data from isotropic materials. To fully understand the idealized condition ofK _I andK _II calculated from thermoelastic experiments, the total stress field calculated from finite element analysis is used to take the place of data obtained from real thermoelastic experiments. In the finite element analysis, theJ-integral is also calculated to compare with (K _I ² +K _II ² )/E evaluated by the least squares method. The stress fields near the crack tip are dominated by the two stress intensity factors; however, the edge effect will cause inaccuracy of the thermoelastic data near the crack tip. Furthermore, the scan area of thermoelastic experiments cannot be too small. Therefore, we suggest that three or four terms of stress function be included in the least squares method for evaluating stress intensity factors via the thermoelastic technique. In the idealized condition, the error can be smaller than 3 percent from our numerical simulations. If only ther ^–1/2 term (K _I andK _II ) is included in the least squares method, even in the idealized case the error can be up to 20 percent.     

Energy principle of ferroelectric ceramics and single domain mechanical model

Many physical experiments have shown that the domain switching in a ferroelectric material is a complicated evolution process of the domain wall with the variation of stress and electric field. According to this mechanism, the volume fraction of the domain switching is introduced in the constitutive law of ferroelectric ceramic and used to study the nonlinear constitutive behavior of ferroelectric body in this paper. The principle of stationary total energy is put forward in which the basic unknown quantities are the displacement u _i , electric displacement D _i and volume fraction ρ _I of the domain switching for the variant I. Mechanical field equation and a new domain switching criterion are obtained from the principle of stationary total energy. The domain switching criterion proposed in this paper is an expansion and development of the energy criterion. On the basis of the domain switching criterion, a set of linear algebraic equations for the volume fraction ρ _I of domain switching is obtained, in which the coefficients of the linear algebraic equations only contain the unknown strain and electric fields. Then a single domain mechanical model is proposed in this paper. The poled ferroelectric specimen is considered as a transversely isotropic single domain. By using the partial experimental results, the hardening relation between the driving force of domain switching and the volume fraction of domain switching can be calibrated. Then the electromechanical response can be calculated on the basis of the calibrated hardening relation. The results involve the electric butterfly shaped curves of axial strain versus axial electric field, the hysteresis loops of electric displacement versus electric filed and the evolution process of the domain switching in the ferroelectric specimens under uniaxial coupled stress and electric field loading. The present theoretic prediction agrees reasonably with the experimental results given by Lynch. The project supported by the National Natural Science Foundation of China (10572138).     

Fracture toughness for CNT specimens from numerically obtained critical CTOD values

Many approaches for estimating mode I fracture toughness (K_IC) using circumferentially notched tensile (CNT) specimen have been demonstrated in the literature. In this paper, an effective approach for estimating fracture toughness from the numerical solution of critical crack tip opening displacement (CTOD) is proposed. An elasto-plastic finite element analysis is used to estimate critical CTOD values for CNT specimens. A number of materials are analysed, and the relationship between K_IC and critical CTOD for CNT specimens is obtained. The proposed relationship is validated by comparing the fracture toughness values obtained from the relationship with those obtained experimentally using CNT specimens. The fracture toughness (K_IC) calculated according to this relationship from numerically obtained critical CTOD is found to be in close agreement with the experimentally obtained fracture toughness for the respective materials.     

A bar impact tester for dynamic fracture testing of ceramics and ceramic composites

A bar impact test was developed to study the dynamic fracture responses of precracked ceramic bars, Al₂O₃ and 15/29-percent volume SiC_w/Al₂O₃. Crack-opening displacement was measured with a laser-interferometric displacement gage and was used to determine the crack velocity and the dynamic stress-intensity factorK _I ^dyn . The crack velocity andK _I ^dyn increased with increasing impact velocity while the dynamic-initiation fracture toughness,K _Id, did not vary consistently with increasing impact velocities.Paper was presented at the 1992 SEM Spring Conference on Experimental Mechanics held in Las Vegas on June 8–11.     

Simultaneous measurements of stress intensity and toughness for fast-running cracks in steel

Simultaneous measurements of the dynamicstress-intensity factorK _I ^dyn and the dynamic-fracture toughnessK _ID were made in a high-strength steel to investigate the relation between energy delivered to and energy absorbed by rapidly propagating cracks. Values ofK _I ^dyn were obtained intermittently during the propagation history by the shadow optical method of caustics from high-speed photographs of the moving crack tips. Values ofK _ID were calculated from temperature maxima recorded by thermocouples near the crack path. The results indicate that for fast-running cracks, the change in energy available at the crack tip can be significantly less than the energy absorbed in crack extension, suggesting that currently used dynamic-energy-balance methods for determining dynamic-fracture toughnesses may provide erroneous values.     

Effect of T-stress on crack growth under mixed mode I–III loading

For a crack subjected to combined mode I and III loading the influence of a T-stress is analyzed, with focus on crack growth. The solid is a ductile metal modelled as elastic–plastic, and the fracture process is represented in terms of a cohesive zone model. The analyzes are carried out for conditions of small scale yielding, with the elastic solution applied as boundary conditions on the outer edge of the region analyzed. For several combinations of the stress intensity factors K_I and K_III and the T-stress crack growth resistance curves are calculated numerically in order to determine the fracture toughness. In all situations it is found that a negative T-stress adds to the fracture toughness, whereas a positive T-stress has rather little effect. For given values of K_I and T the minimum fracture toughness corresponds to K_III = 0.     

Empirical Measures of Wettability in Porous Media and the Relationship between Them Derived From Pore-Scale Modelling

The wettability of a crude oil/brine/rock system is of central importance in determining the oil recovery efficiency of water displacement processes in oil reservoirs. Wettability of a rock sample has traditionally been measured using one of two experimental techniques, viz. the United States Bureau of Mines and Amott tests. The former gives the USBM index, I _USBM, and the latter yields the Amott–Harvey index, I _AH. As there is no well-established theoretical basis for either test, any relationship between the two indices remains unclear.Analytical relationships between I _AH and I _USBM for mixed-wet and fractionally-wet media have been based on a number of simplifying assumptions relating to the underlying pore-scale displacement mechanisms. This simple approach provides some guidelines regarding the influence of the distribution of oil-wet surfaces within the porous medium on I _AH and I _USBM. More detailed insight into the relationship between I _AH and I _USBM is provided by modelling the pore-scale displacement processes in a network of interconnected pores. The effects of pore size distribution, interconnectivity, displacement mechanisms, distribution of volume and of oil-wet pores within the pore space have all been investigated by means of the network model.The results of these analytical calculations and network simulations show that I _AH and I _USBM need not be identical. Moreover, the calculated indices and the relationship between them suggest explanations for some of the trends that appear in experimental data when both I _USBM and I _AH have been reported in the literature for tests with comparable fluids and solids. Such calculations should help with the design of more informative wettability tests in the future.     

The strain energy criterion of mixed-mode brittle fracture

In this paper, we proposed a new cr tenon or mixea-moae brittle fracture, i.e., the strain energy criterion, which can be stated as (K_Ⅰ/K_Ⅰc)² +(K_Ⅰ/K_Ⅰc)²+(K_Ⅱ/K_Ⅱc)²=1. This criterion is shown to be in good agreement with known experimental data.In this paper, an experimental criterion:(K_Ⅰ/K_Ⅰc)^m+(K_Ⅱ/K_Ⅱc)ⁿ=1, 1≤_n^m≤2.is also proposed.     

Dynamic stress-intensity factors for unsymmetric dynamic isochromatics

The mixed mode, near-field state of stresses sourrounding a crack propagating at constant velocity is used to derive a relation between the dynamic stress-intensity factorsK _I,K _II, the remote stress component σ _ox and the dynamic isochromatics. This relation, together with an over-deterministic least-square method, form the basis of a datareduction procedure for extracting dynamic,K _I,K _II and σ _ox from the recorded dynamic photoelastic pattern surrounding a running crack. The overdeterministic least-square method is also used to fit static isochromatics to the numerically generated dynamic isochromatics. The resultant staticK _I,K _II and σ _ox are compared with the corresponding dynamic values and estimats of errors involved in using static analysis to process dynamic isochromatic data are obtained. The data-reduction procedure is then used to evaluate the branching stress-intensity factor associated with crack branching and the mixed-mode stress-intensity factors associated with crack curving.     

Conservation Law and Application of J-Integral in Multi-Materials

The conservation law of J-integral in two-media with a crack paralleling to the interface of the two media was firstly proved by analytical and numerical finite element method. Then a schedule model was established that an interface crack is inserted in four media. According to the J-integral conservation law on multi-media, the energy release ratio of Ⅰ-type crack was considered to be conservation when the middle medium layers are very thin. And the conservation law was also convinced by numerical method. By means of the dimension analysis on the model, the asymptotic results and formula calculating the energy release ratio and complex stress intensity factor are presented.     

A Special Planar Orthotropic Material

This paper shows the existence of a particular type of planar orthotropic material, here denoted for the sake of brevity as R ₀-orthotropic. The number of independent elastic constants for these materials is three, and not four as for a general orthotropic layer, but these constants have only two orthogonal axes of symmetry. The way to obtain a R ₀-orthotropic layer is discussed in the paper, along with the advantages in its use. This revised version was published online in June 2006 with corrections to the Cover Date.     

ON J-INTEGRALS IN THE PLANE FRACTURE OF COMPOSITE MATERIALS

By using a complex function method in this paper, the complex form of J-integral of mixed mode crack tip for unidirectional plate of linear-elastic orthotropic composites is obtained first by substituting crack tip stresses and displacements into general formula of Jintegral. And then, the path-independence of this J-integral is proved. Finally, the computing formula of this J-integral is derived. As special examples, the complex forms, path-independence and computing formulae of J-integrals of mode Ⅰ and mode Ⅱ crack tips for unidirectional plate of linear-elastic orthotropic composites are given.     

Strain-gage methods for measuring the opening-mode stress-intensity factor,K I

Measurements of strain near a crak tip with electrical-resistance strain gages do not usually provide a reliable measure ofK _I because of local yielding, three-dimensional effects and limited regions for strain-gage placement. This paper develops expressions for the strains in a valid region removed from the crack tip, and indicates procedures for locating and orienting the gages to accurately determineK _I from one or more strain-gage readings.